Jordan groups are class of transitive permutation groups (G,Ω) which come with an associated family of subsets of Ω called "Jordan sets". An example is the group of order automorphisms of a dense, linearly ordered set (say (ℚ;<)); in which case the collection of open intervals is a family Jordan sets. By viewing (topologically) closed subgroups of Sym(ℕ) as automorphism groups of first-order structures on a countably infinite set Ω, we use results from the theory of infinite Jordan groups to identify an infinite family of maximally closed subgroups of Sym(ℕ) which contain automorphism groups of certain dense (partially) ordered trees. This work provides countably many confirmations of a model theoretic conjecture of Simon Thomas.

My focus in this talk is to give an exposition of the main concepts and definitions involved (including all those mentioned above).

(During this talk I will also mention joint work with Manuel Bodirsky, Michael Pinsker and András Pongrácz.)

Dehn has introduced three main problems in the algorithmic theory of finitely presented groups: the word problem, the conjugacy problem and the isomorphism problem. The talk discusses these three problems in the special case of polycyclic groups. It shows how the first two problem can be solved for polycyclic groups in general and how the last problem can be solved in the special case of torsion-free nilpotent groups of small Hirsch length.

The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. Morita Frobenius numbers were introduced by Kessar in 2004 in the context of Donovan’s Conjecture in block theory. I will present the latest results of a project in which we aim to calculate the Morita Frobenius numbers of the blocks of quasi-simple finite groups. I will also discuss the importance of a recent result of Bonnafé-Dat-Rouquier for our methods, and explain the relationship between Morita Frobenius numbers and Donovan’s Conjecture.

A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the asymptotic growth of the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

By generalizing the Bestvina-Brady construction I have been able to exhibit an uncountable family of groups of type FP. I shall explain some aspects of the construction and some results that use it, including a proof that every countable group embeds in a group of type FP_2.

Estimation a lower bound on the spectral gap of the Laplace operator on a finitely presented groups (and thus a lower bound on the Kazhdan constant) is a fundamental problem in analytic group theory. In particular, much work was devoted only to establishing and then improving bounds for SL(n,ℤ) or SL(n,𝔽_p). So far the results depend on the property of bounded generation and provide estimation of spectral gaps of the (group) Laplace operator.

Following results of T.Netzer & A.Thom and N.Ozawa we use a different method to estimate the spectral gap. The method is theoretically based on non- commutative Positivstellensatz in real algebraic geometry. The computational side of the method uses mathematical programming (conic optimisation) to provide lower bound on the spectral gap, and thus on the Kazhdan constants. The results are later certified mathematically.

Using this method we were able to improve the existing estimates for groups SL(n,•) (for n=2,3,4,5) by three orders of magnitude.

This is joint work with Piotr Nowak.

Lyndon's theorem that a commutator in a free group is not a proper power, Magnus' Freiheitssatz, Wise's W-cycles conjecture, Stallings' theorem that injections of free groups inducing injections on abelianization are injective on the set of conjugacy classes, and Baumslag's theorem on adjoining roots to subgroups of free groups, are all special cases of a generalized Freiheitssatz. The simplest new case covered resembles the Borromean rings. This is joint work with Henry Wilton.

Let G be a finite group. We define the block graph of G as follows. The vertices are the prime divisors of |G|, and there is an edge between two vertices p ≠ q if and only if Irr(B₀(G)_p) ∩ Irr(B₀(G)_q) ≠ {id_G}; here Irr(B₀(G)_r) denotes the set of complex irreducible characters of G contained in the principal r-block B₀(G)_r.

Motivated by the work of Bessenrodt and Zhang, who proved that block graphs of alternating groups are complete, we investigate block separations of characters in terms of block graphs. We show that block graphs of all finite nonabelian simple groups are complete, except for the sporadic groups J₁ and J₄. Moreover, we determine a criterion of solvability for a group in terms of triangles contained in its block graph. This is a joint work with Brough and Liu.

The module of derivations of multi-hyperplane arrangements was introduced by Ziegler in the 1980's. Maybe the most important example is the Coxeter arrangement with constant multiplicities. This case was solved by Terao in 2002 by showing that the derivation modules are free in this case.

After a detailed presentation of the needed constructions from differential geometry, I plan to present a generalization to all complex reflection arrangements using the most general notion of Coxeter numbers and an operator on irreducible components introduced by Malle in the context of cyclotomic Hecke algebra.

This is joint work with Torsten Hoge, Toshiyuki Mano and Gerhard Roehrle.

Lie algebras and Jordan algebras are two important classes of non-associative algebras. Lie algebras are widely known, because they appear in the theory of Lie groups. Jordan algebras are mainly interesting due to their relations to Lie algebras. To each Jordan algebra J one can associate three Lie algebras embedded into one another: the algebra of inner derivations of J, the structure algebra of J, and the so-called Tits-Kantor-Koecher construction, which is, roughly speaking, the algebra of traceless matrices of order 2 over J. In particular, the simple Lie algebras of types F₄, E₆, E₇ are associated to the exceptional simple 27-dimensional Jordan algebra (the Albert algebra).

The talk will be devoted to yet another construction, which associate a Lie algebra to each semisimple Jordan algebra of (generalized) degree 3. This Lie algebra contains the Tits-Kantor-Koecher construction as a proper subalgebra. In this way one can uniformly obtain all the simple Lie algebras but A_n and C_n, together with some natural embeddings between them.

A finite group is called rational if all entries of its character table are rational integers. Being rational has significant implications for the structure of the group, e.g. it is a classical result of R. Gow that the only primes dividing the order of such a group are 2, 3 and 5, if the group is solvable. The concept of rationality was generalized in 2010 by D. Chillag and S. Dolfi by introducing the term (inverse) semi-rational group. It turned out that being an inverse semi-rational group has quite some impact in the study of integral group rings. We will discuss this connection and recent results.

We look at a parabolic subgroup P of GL_n which acts on the variety of nilpotent matrices in its Lie algebra. Our main question is: is there only a finite number of orbits or do we find infinitely many of them? We translate the group action to a setup in the representation theory of a finite-dimensional algebra and make use of different methods (for example Auslander-Reiten Theory, known infinite families of minimal tame algebras, etc.). This way, we find a complete finiteness criterion which classifies every parabolic subgroup P which acts with only a finite number of orbits.

Let G be a semisimple group defined over a non-Archimedean local field K, typically SLn(ℚ_p), with residual characteristic p. We have a topological group structure on the group of rational points G(K). The maximal compact subgroups of G(K) can be realized as some stabilizers for a suitable action of the group G(K) on a polysimplicial complex called the Bruhat-Tits building. In this presentation, we begin with an short introduction to the theory of Bruhat-Tits buildings, which is used to describe the maximal compact and pro-p subgroups of G(K). The latter play a role analogous to that of the p-Sylows of a finite group and are, in particular, pairwise conjugated. Under suitable assumptions, we can then explicitly describe a minimal set of topological generators of a maximal pro-p subgroup. The minimal number of these topological generators is then linear in some combinatorial datum defined from G, namely the rank of a well-chosen root system.

Many challenging instances of isomorphism in algebra concern determining if two tensors are equivalent under a change of basis. Unlike in linear algebra, data in multilinear algebra involves problems from algebraic geometry, wild representation theory, and solving NP-complete problems. Towards a better approach, we introduce a correspondence between tensors, polynomials, and operators and follow its consequences. We report on joint work with Uriya First and James B. Wilson.